LP INEQUALITIES FOR THE PRODUCT OF THE SUPREMA OF SEVERAL MARTINGALES STOPPED AT RANDOM TIMES - WEIGHTED NORM INEQUALITIES

Let (M(t), t greater-than-or-equal-to 0) and (N(t), t greater-than-or-equal-to 0) be two continuous martingales defined on a filtered probability space (OMEGA, (F(t), t greater-than-or-equal-to 0), P). We suppose that M0 = N0 = 0. We discuss different extensions of the Burkholder-Gundy inequalities, when, instead of [GRAPHICS] and E[[M]T(p/2), we consider such quantities as [GRAPHICS] and E[[M[K(p/2) [N]L(q/2)]. Here K and L are stopping ping times or any positive F(infinity)-mesurable variables. We also obtain some weighted norm inequalities, that is Burkholder-Gundy like inequalities under another probability measure which is absolutely continuous with respect to P.